---
title: "Clos network and STS/TST blocking"
subject: "Telecommunications"
chapter: 4
key: "semester_8/telecommunications:clos-network-sts-tst-blocking"
source: https://www.bctengineer.com/q/semester_8/telecommunications/clos-network-sts-tst-blocking
site: BCT Engineering
---

# Clos network and STS/TST blocking

Telecommunications · Chapter 4: Digital Switching

- Asked in: 2081 Chaitra, 2+3+3 marks
- Video: [Three stage Networks](https://www.youtube.com/watch?v=1RnBCgqJkjw) — @Ahmad Sir 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/telecommunications/clos-network-sts-tst-blocking
- Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-4

## Question

Define what the Clos network is. Derive the expression for blocking probability of both STS and TST switches, and compare them.

## Short answer

Cls network is a three-stage switch: N inputs divided into N/n first-stage n×k arrays, k middle (N/n)×(N/n) arrays, N/n third-stage k×n arrays; strictly non-blocking if k = 2n-1. Blocking probability: STS: P_B = [1 - (1 - p/beta)^2]^k, beta = k/N; TST: P_B = [1 - (1 - p/alpha)^2]^l, alpha = l/c. TST blocking is far lower because l >> k.

## Answer

#### Clos network
A **Clos network** is a three-stage (multi-stage) switching network, proposed by Charles Clos (1953), in which $N$ inputs are divided into $N/n$ first-stage arrays of size $n \times k$, connected to $k$ middle arrays of size $\frac{N}{n} \times \frac{N}{n}$, which connect to $N/n$ third-stage arrays of size $k \times n$. Every array of one stage has exactly one link to every array of the next stage.
$$
\begin{aligned}
N_x &= \underbrace{\tfrac{N}{n}(nk)}_{\text{stage 1}} + \underbrace{k\left(\tfrac{N}{n}\right)^2}_{\text{stage 2}} + \underbrace{\tfrac{N}{n}(kn)}_{\text{stage 3}} \\
&= 2Nk + k\left(\frac{N}{n}\right)^2
\end{aligned}
$$
Clos showed that the network is **strictly non-blocking** when $k = 2n - 1$ (worst case: $n-1$ other inputs of the calling array and $n-1$ outputs of the called array use different middle arrays, so one more is needed). With the best choice $n \approx \sqrt{N/2}$, the minimum is $N_x \approx 4N(\sqrt{2N} - 1)$, much less than $N^2$.

#### Blocking probability of STS
```text
 STS:     S (N x k) -> T (k TSI modules) -> S (k x N)
 Lee graph: k parallel paths, each = 2 links in series
```
In STS, a call from input highway A to output highway B in given slots can go through any of the $k$ time-switch modules. Each path uses a link from the first space stage to a module and a link from the module to the second space stage. With link occupancy $p' = p/\beta$ ($\beta = k/N$ space expansion):
- Probability a link is free $= 1 - p'$; both links of a path free $= (1-p')^2$.
- Path blocked $= 1 - (1-p')^2$; all $k$ paths blocked:
$$
\begin{aligned}
p' &= \frac{p}{\beta},\quad \beta = \frac{k}{N} \\
P_B(\text{STS}) &= \left[1 - (1-p')^2\right]^{k}
\end{aligned}
$$

#### Blocking probability of TST
```text
 TST:     T (c -> l slots) -> S (N x N) -> T (l -> c)
 Lee graph: l parallel paths (internal time slots)
```
In TST, the path from A to B passes through the space stage in any of the $l$ internal time slots. Each path uses one internal slot on A's highway and the same slot on B's highway. With time expansion $\alpha = l/c$, $p' = p/\alpha$, and by the same argument:
$$
\begin{aligned}
p' &= \frac{p}{\alpha},\quad \alpha = \frac{l}{c} \\
P_B(\text{TST}) &= \left[1 - (1-p')^2\right]^{l}
\end{aligned}
$$

#### Comparison
Since the number of paths in TST equals the number of internal time slots $l$ (32 to several hundred), while in STS it equals the number of time-switch modules $k$ (usually small), TST blocking is far lower for the same hardware.
Example with $p' = 0.2$:
$$
\begin{aligned}
\text{STS},\ k = 8 &: \ (1-0.8^2)^{8} = 0.36^{8} \approx 2.8 \times 10^{-4} \\
\text{TST},\ l = 32 &: \ (1-0.8^2)^{32} = 0.36^{32} \approx 6.3 \times 10^{-15}
\end{aligned}
$$

| Point | STS | TST |
|---|---|---|
| Number of alternative paths | $k$ (time modules) | $l$ (internal slots) |
| Blocking | Higher | Much lower |
| Expansion by | More space arrays/modules | More internal slots (cheap memory) |
| Cost for large exchange | Higher | Lower (time stages cheap) |
| Control | Simpler for small sizes | More memory, path search over slots |
| Use | Small exchanges | Most large digital exchanges |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A Clos three-stage network with n inputs per first-stage array is strictly non-blocking when the number of middle arrays k is:
   A. n
   B. 2n - 1
   C. 2n + 1
   D. n^2
   Answer: B. Worst case n-1 other inputs and n-1 other outputs use different middle arrays, so one more is needed: k = 2n-1. (Key point: Clos non-blocking condition)

2. What is the total crosspoint count of a Clos network?
   A. N^2
   B. Nk + k(N/n)
   C. 2Nk + k(N/n)^2
   D. k(N/n)^2 only
   Answer: C. Stage 1 plus stage 3 give 2Nk, and the k middle arrays of size (N/n)x(N/n) give k(N/n)^2. (Key point: Crosspoint formula)

3. In the Lee graph of STS, how many parallel paths exist between input and output?
   A. l internal slots
   B. The number of crosspoints
   C. k time-switch modules
   D. Exactly one
   Answer: C. A call can use any of the k time-switch modules; TST instead has l internal slots. (Key point: Number of paths in STS)

4. Why is TST blocking much lower than STS for the same hardware?
   A. TST has fewer links
   B. TST has no space stage
   C. STS has more internal slots
   D. TST has l paths (many internal slots) against few modules k in STS
   Answer: D. Blocking is [1-(1-p')^2] raised to the number of paths; l is far larger than k, so TST blocks far less. (Key point: TST versus STS blocking)

## Related in Telecommunications

- [Derive STS and TST blocking probabilities](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Asts-tst-blocking-derivation) — Telecommunications, chapter 4
- [Space and time switches; STS vs TST](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aspace-time-switch-sts-tst) — Telecommunications, chapter 4
- [Multistage switching; TST vs STS](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Amultistage-tst-sts-compare) — Telecommunications, chapter 4
- [TST time slots for non-blocking operation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atst-time-slots-numerical) — Telecommunications, chapter 4
- [Short note: TST and STS](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Ashort-note-tst-sts) — Telecommunications, chapter 4
- [Telephone exchange structure; TST vs STS](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aexchange-structure-tst-sts) — Telecommunications, chapter 4